Show that, in a group $G, \circ(ab) = \circ(ba)$ for all $a,b \in G$ . Further show that if $a$ and $b$ commute, and $\circ(a) =n_1 , \circ(b) =n_2$ and $(n_1,n_2) =1$, then $\circ(ab)= \circ(ba)=n_1 n_2$ .

Let $G$ be a group, $H \le G$ and $g \in G$ . Show that if $\circ(g) =n$ and $g^m \in H$, where $n$ and $m$ are coprime integers, then $g \in H$ .

Let $G$ be a group, $H \le G$ and $g \in G$. Show that if $\circ(g) =n_1n_2$ and $g^m \in H$, where $n$ and $m$ are coprime integers, then $g \in H$ .

Let $G$ be a group and $g \in G$ with $\circ(g) =n_1n_2$ , where $n_1$ and $n_2$ are coprime positive integers. Then show that there are elements $g_1,g_2 \in G$ such that $g = g_1g_2 =g_2g_1$ and $\circ(g_1) =n_1 , \circ(g_2) =n_2$ . Further show that $g_1$ and $g_2$ are uniquely determined by these conditions.

Let $G$ be a group such that the intersection of all its subgroups which are different from $\{e\}$ is a subgroup different from $\{e\}$ . Prove that every element in $G$ has finite order.

If a group $G \neq \{e\}$ has no nontrivial subgroups, show that $G$ must be a finite group of prime order.

Give an example of a group $G$ having a subgroup $H$ and two elements $a,b$ such that $Ha=Hb$ but $aH \neq bH$ .

Let $H \le G$, and $a \in G$ . Let $aHa^{-1} = \{aha^{-1} |h \in H \}$ . Show that $aHa^{-1}$ is a subgroup of $G$.

Suppose that $H \le G$ such that whenever $Ha \neq Hb$ then $aH \neq bH$ . Prove that $gHg^{-1} \subseteq H$ for all $g \in G$ .

If $H$ and $K$ are subgroups of orders $p$ and $n$ , respectively, where $p$ is a prime number, show that either $H \cap K=\{e\}$ or $H \le K$ .

If $G$ is a group and $K \le H \le G$ , then show that $[G:K] =[G:H][H:K]$ (assume finite indices). Hence deduce that if $[G:H] =p$, where $p$ is a prime, then either $H=K$ or $H=G$ .